5 Answers
5

All Cohanim and Levi'im are descended from the tribe of Levi. During the time of the temple, a Cohen would have to provide documentary evidence of their ancestry in order to serve in the temple (כהן מיוחס). Since the Babylonian exile, all such documentary evidence has been lost (with the exception of perhaps the Rappaport family). Some interesting genetic studies have also been done relating to the shared ancestry of Cohanim. Today halachically, anyone who claims to have a tradition as being a Cohen or Levi is accepted as one, for the sake of the privileges such a role entails (like getting the first or second aliyah, being able to give the priestly blessing).

Aside from Cohanim and Levi'im, there are many people who have family traditions that relate them back to Kind David (placing them in the tribe of Judah). These usually consist of direct family trees going back to a Jewish sage (rishon or achron) who lived some time in the last thousand years, and who have been related back to King David (for example, Rashi and the others).

Aside from that, no one can claim with even a small degree of certainty to which tribe they are descended from (assuming patrilineal descent). After the exile of the "10 Lost Tribes" into the Assyrian empire, there were either 3 or 4 tribes left - Levi, Judah, Binyamin and probably Shimon. Most today are descended from one of the last three (with the greatest likelihood of descent from Judah, as it had the biggest population at the time of the first exile).

In addition to what @Danny answered (Cohanim and Levites are from the tribe of Levi), several families claim to trace their lineage back to King David, meaning they are from the tribe of Judah.

Since the destruction of the Northern Kingdom of Israel during the time of the First Temple, 10 tribes were lost and we have no trace of them. The only tribes that were not lost were Judah and Benjamin (and the Levites and Cohanim who lived in their territories). Benjamin was very small numerically, compared to Judah, so pretty much every Jew alive today who is not a Cohen or Levi is probably from the tribe of Judah.

All that said, it's very difficult to "prove" any such genealogical claims; anybody could invent a family tree proving their descent from pretty much anyone they choose. We really just rely on people's good faith when they claim to be a Cohen or Levi, but there are practical halachic rulings that effectively acknowledge the difficulty in proving genealogy. For example maaser rishon (the first tithe, 10% of produce grown in Israel) should be given to a Levi - but since nobody can prove beyond doubt that they are a Levi, the obligation to give it to a Levi is unenforceable, and the owner himself may keep it.

The "10 tribes" may not have really been 10. See here - Shimon's territory was completely surrounded by Judah, so was most likely part of the Kingdom of Judea (in which case, Shimon, Levi, Yehudah and Binyamin were not part of the "lost" tribes).
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Yaakov EllisMay 11 '11 at 11:21

I think that it does affect your point, since Shimon would now be added to the potential tribes from which someone could be descended (lowering the statistical chance that a non-Levi would be from Judah)
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Yaakov EllisMay 11 '11 at 11:28

Note that some who claim to descend from David or the like do not claim to do so via a male line, so that wouldn't affect what tribe they're from.
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msh210♦May 11 '11 at 15:48

There is a chassidic story (I've seen it in A Treasury of Chassidic Tales by R' S.Y. Zevin; online here) in which R' Tzvi Elimelech of Dinov is told by his rebbe, the Chozeh of Lublin, that he descends from the tribe of Yissachar. Indeed, R' Tzvi Elimelech named his major work "Bnei Yissaschar" in commemoration of this.

So there's a rare case of a family with evidence of descent from a particular tribe. (Though it is possible that the Chozeh meant that R' Tzvi Elimelech's soul is associated with Yissachar, not necessarily that he was biologically descended from him.)